FAMILIES OF ALGEBRAIC TEST EQUATIONS W. GAUTSCHI (I)

Transcription

1 FAMILIES OF ALGEBRAIC TEST EQUATIONS W. GAUTSCHI (I) 1. Introduction. Test matrices have been in use for some time to scrutinize computer algorithms for solving linear algebraic systems and eigenvalue problems; see, e. g., Gregory and Karney [1969]. For the problem of finding roots of algebraic equations, the construction of appropriate test equations has been given less attention. Here we wish to propose two families of algebraic test equations, the first consisting of equations with predominantly complex roots, the second of equations with only real roots. To be useful for testing purposes, a test equation (of some fixed degree) should have the following characteristics: (1) All roots of the equation can be calculated directly (i. e., without recourse to a rootfinding algorithm). (2) The equation contains a parameter (or parameters) which can be used to control the numerical condition of the roots. By varying the parameter (s), the condition number of the worst-conditioned root can be made to range from relatively small to arbitrarily large values. (3) All coefficients of the equation are integer-valued. It may not be easy, in practice, to achieve all these characteristics, particularly the last one, if we are interested in relatively large degrees. Even when this is possible, the integer coefficients may become so large as to make exact representation in floating-point arithmetic impossible. Although equations which do not satisfy (3) are less desirable, they are still useful for testing purposes, Received ]une 6, ) Department of Computer Sciences, Purdue University, Lafayette, Indiana 47907, U.S.A..

2 384 W, GAUTSCHI: Families provided one takes properly into account the influence of rounding errors in the coefficients upon the results. Isolated examples of test equations, particularly ill-conditioned ones, have been known for a long time. Perhaps the best-known example, due to Wilkinson [1963], is the equation with roots at 1, 2... n. Some of these roots are relatively well-conditioned, while others are quite ill-conditioned, more so the larger the degree n. (The numerical condition of Wilkinson's equation is analyzed in Gautschi [1973]; see also Gautschi [1978, w 4]). The roots of unity lead to another interesting example if one removes half of them and retains only those on the half-circle (lenkins and Traub [1975]). Our first family of test equations, indeed, is a simple extension of this latter example. 2. A first family of test equations and their numerical condition. Given an algebraic equation of degree n, (2.1) p (z)=0, p (z)=z~+al z~-l+...+a,,_l z+a,, with complex coefficients ax, and a simple root ~ of (2.1), we adopt as condition number for ~ the quantity (Wilkinson [1963, p. 38 if], Gautschi [1973]) (2.2) cond ~-- z rail Ir ]p' ( )l " It measures the sensitivity of ~, in terms of relative errors, to small (relative) perturbations in all nonzero coefficients az. Our first family of test equations (with parameter a) is (2.3) p~ (z) =0, p, (z)= /-) [z-r (a)]=z"-~ z"-l+...+(-1)" ~,, 2~ r (co =e(~-')% O<~z_< -- n If ~z=21r/n, the ~ are the n-th roots of unity, thus p~ (z)=z ~- 1, and we are in the case of a well-conditioned equation, all roots having condition 1/n. For a=rr/(n--1), we get the roots of unity on a half-circle, which are all relatively ill-conditioned. As c~ $ 0, the condition deteriorates unboundedly,

3 o~ algebraic test equations 385 From Eq. (2.2) we find for the condition of the roots in (2.3), (2.4) cond (~= ~=', v=l,2... n. a,~, (Z 2 ~'-I H sin (v--2) -~- In view of the symmetry property cond~=cond~+l-~, v=l,2... n, it suffices to consider the condition numbers for v= 1, 2... n', where We show that In + 1] n'=[ 2 I" 27~ (2.5) cond ~1< cond ~2<...< cond ~, for 0<a< --. ll If n=2, there is nothing to prove. If n>2, we let zr~ denote the product in the denominator of (2.4) and observe that (2.6) ~v-1 sin (v-- 1) 2 sin (n--v+ 1) 2 v=2, 3... n'. Our assumption on a implies that both sine functions in the numerator and denominator are evaluated in the open interval (0, re), the former in fact in (0, re/2). The absolute value signs in (2.6) can thus be deleted, and an elementary calculation shows that all ratios in (2.6) are less than 1 precisely if ~Z ) 6~ /,/,. (2.7) tan ~t~- > tan(v--l)-~-, v=2,3... Since the tangent is monotone increasing on [0, z~/2), the inequality (2.7) for v = n" implies all others, and indeed holds true by virtue of,(o Itan n 9 if n is even, a _. 4 2 tan (n'-- 1) -- tan n 4 4 if n is odd. It follows that zh>z~2>...>~r,., hence (2.5).

4 386 W. GAUTSCHI: Families We recall that the condition number in (2.2) is invariant with respect to scaling of the independent variable. Any transformation z=coz*, where co~=0 is arbitrary complex, leaves the condition of the roots unchanged. As a consequence, the assumption ]~,] = 1 made in (2.3) does not restrict generality, and we are free to subject the roots to a rigid rotation on the unit circle. Taking advantage of this last remark, we may bring the equation p~ (z)=0, which generally has complex coefficients, into a form with real coefficients. It suffices to rotate the roots into a position symmetric with respect to the real axis, i. e., to put Then 2 ~ e i(n-1)a/2 Z*. (2.8) p~ (e ic~-1~/2 Z*) "-- e i"(n-ll~'/z p,* (z*), where p~* (u)= /~ [u-~* (col,. x i(2--1--n--l~a ~* ta)=e T/, ~=1 and ~x*=~*.+l-~, 2=1,2... n, implying that all coefficients of p~* indeed are real. 3. Computation of the coefficients of p~* and the condition numbers cond ~. In analogy to (2.3), we write (3.1) p~*(u)= /7 [u--~*(a)]= 21 (--1)~*u ~-z, ~ro*=l, ).=1 ~0 where all crx* are real. Observing from (2.8) that p~ (z)--e incn-1)a/2 p~* (e -i(n-1)~/2 z), and using (3.1), we find i p~ (z)= 21 (--1) 4 e T M a~* z n-~. Hence, by comparison with (2.3), (3.2) az=e T M az*, 2=0, 1, 2... n.

5 of algebraic test equations For the following, it will be necessary to notation a2--o'4.., indeed, to define ~r~., by 387 introduce the more precise (3,3) /7 [z-r (a)] = Z (-1) 4 o'4,, z "-z, 1=I 4=0 /z----l, 2, 3... and similarly ~r*a,~ by (3.4) 1-1 [u-~*4,;, (~z)] -- X (-1) ~ a*4. u "-4, I=1 a~o where /z=l,2, 3... r a,f, (~z) = e i (4-1- e_~)=, I -- 1, 2... l.z. Then, as in (3.2), (3.5) O'2,tt = e T M a*2,/~ 9 Defining (3.6) a0,,=l, ~+x,~=0 for /z=0,1,2... one obtains from (3.3) the well-known recursion (3.7) a~,~+,=aa,~+ ~+i o'4_,,~, 2=1,2... /z+l, where ~+~=e ~=. Substituting (3.5) into (3.6), (3.7), and observing that all a*a,~ are real, one finds 0"*0,~ -" 1, O'*t,+l,~-- 0, o*z,u+~= cos (2a/2) a*z,u+ cos ((/z--2+ 1)a/2) a*z-l,~ ;z=0. 1, ,2... /z+l A simple induction argument will show that a*z,u--a*u_z., 2=0, 1,2... /.t. It suffices, therefore, to compute a*o,u-- 1, /z---0, 1... n-- 1,

6 388 W. GAUTSCHI: Families ~*x,~+l: cos (~.a/2) a*~,.+ cos ((#--Z + 1) a/2) G*~_,,.' 1 (5.8),l = 1, 2... ~ ' ~ = 0,1... n- 1. a~ =o",,* -1 ~-i,~,+1 7' **-1- (if /z is even) 1 / The numerator in the condition number (2.4), by (5.2), can now be obtained from ~-1 2 i1+ Ion,, +2 Z I~ a,,j, n even, -- 0"* "--" z t<,nl ~.=1 4=1 n Z [~z%,.[, n odd. In Figure 1 we show the largest condition number, cond r of a--2ztx/n, O_<x_<l, for n=10,20,40. log,o(cond) as function n =40 20 n - 20,, v, I,,, i I I I ~ ~"~'~'I, I ~ X Fig. 1. Condition number of the worst-conditioned root of p~ (z)=0, ~z=~.z~,xln, for 0<x<l, n- 10, 20, 40.

7 o/ algebraic test equations Equations with integer coefficients. The polynomials p~*, as constructed in Section 3, will not have integer coefficients, in general. It is desirable to find a subset of parameter values a, and a suitable constant multiple p~** of p~* such that all coefficients o-~:** of p~** become integer-valued when a is restricted to the subset in question. We can achieve this by letting a 1 (4.1) cos~-=l----, p>0. P Then all cosine factors in (3.8), hence all a*~,,, become polynomials in p-1 with f x f ~. integer coefficients, since cos(m2)=tm(1--~-), m=1,2,3... where T,~ is the Chebyshev polynomial of the first kind. Multiplying through by an appropriate power of p we obtain a polynomial p~** whose coefficients cry,,** are polynomials in p with integer coefficients, hence integer-valued if p is an integer. Selecting p sufficiently large we can get a arbitrarily small, and thus produce equations which are arbitrarily ill-conditioned. The only serious limitation of this procedure is due to the finiteness of the machine arithmetic in which the integer p and, above all, the resulting integer coefficients, are to be represented. A lower limit on p is imposed by the condition a<_2rc/n, which translates into (4.2) 1 p>_ sin 2 L 2n In Table 1 the results are summarized for 2_<n_<8. Only the coefficients a~:**, 2<_ [n/2], are listed, as the others can be obtained by symmetry, a2.,n** = -- 0"* :~ O" ** O" ** O" ** ** n 0,n l,n 2,n O'3,n* * 0"4, n 2 p s 2 (p) 3 1r 5 (P) 1# 2ps2 (P) s 4 (P) 2s 3 (P) s 4 (P) 5 # r 2 s 5 (p) 2s4 (p) ss (p) 6 p9 p4 s2 (p) S6 (p) ps 5 (p) s6 (p) 7 p12 p5 S7 (p) /)2 S6 (p) S7 (p) 8 p16 4p 9 S 2 (P) S 8 (p) 41# S 7 (p) S 8 (P) 2s2 (p) s4 (p) s5 (p) ~ (p) s5 (p) s7 (p) ~ (p) 4ps 2 (P) s7 (P) s 8 (P) s6 (P) 2ss (p) s7 (p) ~ (p) s-[(p) TABLE l. The coefficients of the test polynomial p=** (u)= j~ (--1) z cry,,*** u n-z. ~.~0

8 390 W. OAUTSCHI: Families The notations used in Table 1 are as follows: s2 (p) : 2 (p -- 1) sa (p)= (p--2) (3p--2) s4 (p)=p2--4p+2 s5 (p) = 5p 4-40p p2_ 64p + 16 s6 (p) = 3p ~- 32p p 2-64p + 16 s7 (p) = 7p 6-112p p 4-960ff + 880p 2-384p + 64 ss (p) = p6_ 20pS + 106p 4-224p p z- 96p + 16 s6 (p) = p2_ 8p + 4 ss (p) -- p4_ 16p p-'-- 32p + 8. Inequality (4.2) imposes the constraints (4.3) p>_l, p_>2, p_>4, p_>6, p~8, p_>ll, p_>14 for n = 2, , respectively. Thus. for example, the family of test equations of degree 5, according to Table 1 (n=5), is p6 z 5_ p2 ss (p) z4+ 2s4 (p) S5 (p) Z 3-2S4 (p) S5 (p) z 2-1- p2 S5 (p) Z- if= O, p > Numerical tests. Test equations can be put to many uses. One of the more important ones is the comparative analysis of different algorithms (and their computer implementations) for solving equations. Since we can vary the numerical condition of the roots, for each fixed degree, we are able to observe the performance of these algorithms on equations of fixed degree under increasing pressures of illconditioning. Our main interest, here, being in the accuracy of the algorithms, it is interesting to observe how closely the relative errors in the results are going to conform with what one ideally expects, namely that all these errors are of

9 oj algebraic test equations 391 the order of the machine precision multiplied by the respective condition numbers. If ~+0 denotes an exact root of the equation, "~ the approximation to returned by the computer algorithm, and e denotes the machine precision (i. e., ~=2 -t, where t is the number of binary digits in the mantissa of the floatingpoint word), then for a good algorithm the quantity (,5.1) /z (~)=E I~[ cond ought to be of the order of magnitude 1, or even smaller (considering that cond is a somewhat conservative measure of the condition of ~). The larger # (~), the poorer the performance of the algorithm (with regard to accuracy). If we are interested in the performance of the algorithm on an equation, rather than an isolated root, we can measure it by the average (5.2) z (r H k=l where n is the degree of the equation and ~k its n (simple) roots. By way of illustration, we compare two subroutines, ZRPOLY and ZPOLR, cut cf the IMSL library (International Mathematical Statistical Libraries, Version 7). The first implements the three-stage algorithm of Jenkins and Traub [1970], while the other is based on Laguerre's method, as implemented by Smith [1967]. We compiled both subroutines on the CDC 6500 computer, using the FTN compiler (CDC Fortran Extended Compiler), and ran them on the test equations p~*=0 of degrees n=5,10,20,40, with a=2zrx/n, x=.2, The coefficients a*z.~ were generated by (3.8) in double precision, and then rounded to single precision. Likewise, we used double precision to compute reference values for the roots ~z* (a). The results are summarized in Table 2 (2). The last two columns exhibit, for ZRPOLY and ZPOLR, respectively, the values observed for the average measure/za~ in (5.2). The two preceding columns contain the smallest and the largest condition number associated with the n roots in question. It is seen that the performance of both algorithms is quite good, in general, even on ill-conditioned equations. ZPOLR yields consistently smaller values of/z,~ (with one exception, for n-'5, x=.6). ZRPOLY seems to have some difficulties maintaining accuracy when x is near 1.0 and n is large (i. e., for relatively well-conditioned equations!). (2) Numbers in parentheses indicate decimal exponents.

10 392 W. GAUTSCHI: Families TABLE 2. Performance of ZRPOLY and ZPOLR on the test equation p~*=0, ~z=2rcx/n, x=.2(.2)1., n=5, 10,20,40. n x min cond max cond Na~ ZRPOLY Nay ZPOLR (3).184 (4) (--1) (2).960 (2) (--1) (1).135 (2) (i).229 (1).750 (2) (2) (6).408 (8) (3).564 (5) (--1) (2).761 (3) (1).176 (2).845 (2) (2).200 (1) (12).436 (17) (--1) (7).412 (11) (4).489 (7) (I).188 (4).687 (1) (--1).500 (--1).409 (5).100 (1) (25).973 (55).185 (--11).161 (--11) (13).451 (25) (--2) (7).599 (15).225 (1).707 (--1) (3).426 (8).200 (3).804 (--1) (--1).250 (--1).554 (I0).200 (1) Interestingly, both routines do not fall to pieces when confronted with an extremely ill-conditioned equation, such as the one for n=40, x=.2. All roots are obtained with relative errors ranging between 30 and 120%. This accounts for the very small values of /~a~ shown in this case. Similar results (3) are observed on equations with integer coefficients (those of Table 1), where n--2, , and where p is varied between the smallest 9 admissible integer (cf. (4.3)) and the largest possible integer subject to exact representation of the coefficients ~,n** in CDC 6500 floating-point arithmetic. (a) One of our tests runs (for n=3, p=8xl07) uncovered an implementation error in one of the auxiliary routines called by ZRPOLY.

11 o] algebraic test equations A second family of test equations. We now briefly describe our second family of test equations, viz., (6.1) q~ (z) =0, q~ (z)= /7 [z--lz (a)] =z~--vl z~-l+...+( - 1)" v,, ~.~1 ~(a)=a -~, l<a<co. The equations (6.1) are well-conditioned, when a is large, and become progressively more ill-conditioned as a approaches 1. The condition number of g~, using Gautschi [1973, Thm. 3.1], is easily found to be (6.2) where 27"g+v_l Tg+n_v -- o~-v(v-1)/2 cond{~=, v=l,2...,n, ~-v-1 ~-n-v zr0-*-"l, ~zu // (l_a-z), /x=l,2... n. It follows that while Furthermore, cond fl---~ 1, condf~--~2 (v>l), as a--~ oo, cond~--~ as as1. ~-4 v--1 7~+tn-v cond~<2, v=l,2... n. ~'-v-i ~-n-v The bounds on the right are symmetric, i. e., invariant under the substitution v--~ n+ 1-v, and increase monotonically for v<n/2, attaining their maximum value at v = [n/2] + 1. The true values of the condition number behave similarly. We have, in particular, ["/-~] (1 +,a -z ~ i6.3) J~,<,~max cond ~ < ~=1/7 \~-X ] 9 The largest condition number is shown in Figure 2 as a function of a, for n=10,20,40.

12 .594 W. GAUTSCHI: Families log.,(cond) t t a Fig. 2. Condition number of the worst-conditioned root of %(z)=o, for o~1, n= 10, 20, 40. The coefficients v~=va,n in (6.1) are easily generated by recursion, as in (3.6), (3.7), (6.4) ~-0,g=l, v~+l,u=o, /z-----o, 1,2... 7:a,g+l=vx,u+.ff~+l T~-1,~, 2=1,2... /z+i, /z=o, 1,2... n--1. If we let 1 ~=1+--, p>o, P and multiply through by appropriate powers of p-f 1, the new coefficients z*a,n become polynomials in p with integer coefficients, hence integer-valued, if p is an integer. More specifically, they assume the form

13 oj algebraic test equations (1-4-1) (n--~.) (n+l--l~ (6.5) ~*~,.=p ~ (p+ 1) 2 "r~..** (p), 2=0, 1,2... n, where v~,.** are polynomials satisfying zo,~** (p) = 1, v~..** (p)='c**._~..(p),,~=0, 1,2... n. The polynomials v~,.**(p). 2=1,2... In/2], for 2_<n_<_8, are Table 3. shown in TABLE 3. The coefficients of the test polynomial q~* (z)= j~ (--1).~p),(~+l)/2 (p.jrl)(n-2)(n*l-2)/2 T).,n** (P) zn-2,l.~0 n Tl,n** T2,n** T3,n** T4,n** 2 h (P) 3 t 3 (p) 4 t 2 (p) t 4 (p) t 3 (p) t 4 (p) 5 t 5 (p) t 4 (p) t 5 (p) 6 t 2 (p) fi (p) t~ (p) t 3 (p) t~ (p) t 6 (p) 7 6 (P) ta (P) t6 (P) t7 (P) 8 t 2 (p) t 4 (p) t 8 (p) t 4 (p) t 7 (p) t 8 (p) t 2 (P) t4 (P) t 5 (P) t 6 (P) t5 (P) t6 (P) h (P) h (P) t~ (p) t 6 (p) t 7 (p) t 8 (p) t5 (p) t6 (p) t7 (p) ts (p) The notations used in Table 5 are as follows: h (p) =2p+ 1 h (p)=sp2+sp+ 1 t4 (p)=2p2+ 2p-t - 1 (p)=5r lotr loy+ 5p+ 1 t6 (p)=p2 +p+ 1 t7 (p)= 7p6 + 21pS + 35p4 + 35p3 + 21pZ + 7p+ 1 t8 (p)=2pa +4p3 +6p2 +4p+ 1.

14 596 W. GauTscm: Families Thus, for example, the family of test equations of degree 5, according to Table 3 (n = 5), is (p-j- 1) 15 zs--p (pq- 1) l~ ts (p) Z4q'-p 3 (pq- 1) 6 t4 (p) ts (p) z 3- --ff (p+ 1) 3 t4 (p) ts (p) z2+p ~~ (p+ 1) t5 (p) z--p's=o, p--> Numerical tests. The same two subroutines, as in Section 5, were tested on the equations q~*=0 of degrees n=5, 10,20,40, for selected values of a between 1 and 5. The results are reported in Table 4, in an outlay similar to the one in Table 2. On the whole, the subroutines perform similarly as on the first test equations, ZPOLR generally yielding smaller values of /xav, but the differences are not as pronounced as before. In the case n=40, a=5, a series of low-order coefficients underflows, making a meaningful test impossible.

15 of algebraic test equations 397 TABLE 4. Performance of ZRPOLY and ZPOLR on the test equation q~*=0,,, =1.05 (var.) 5.0, n=5, 10,20,40. n ~ min cond max cond g~. ZRPOLY g~ ZPOLR (6).137 (7).256 (2) (5).947 (5).977 (--1).564 (--1) (3).326 (4) (2).329 (3) (--1) (2).491 (2) (--1) (1).527 (1) (10).231 (12).381 (1).450 (--1) (7).584 (9).148 (1) (4).363 (6) (3).333 (4) (2).113 (3) (1).549 (1) (15).759 (19).231 (--2) go (10).324 (14) (5).206 (8) (3).105 (5) (2).136 (3) (1).550 (1) (19).148 (29).133 (--5).758 (--2) (--1) t.1.i87 (11).889 (I8) (--1) (5).116 (9) (3).124 (5) (2).136 (3) (1).550 (1) (--6)

16 398 W. GAUTSCHI: Families REFERENCES YV'. GAUTSCHI: On the condition o/ algebraic equations, Numer. Math. 21, (1973), W. GAUTSCm: Questions o~ numerical condition related to polynomials, in: <<Recent Advances in Numerical Analysis >~ (1978), C. de Boor and G. H. Golub, eds., Academic Press, New York, R. T. GREGORY and D. L. KARNEY: A Collection o~ Matrices ]or Testing Computational. Algorithms (1969), Wiley-Interscience, New York. ['Corrected reprint, R. E. Krieger Publ. Co., Huntington, N. Y., 1978]. M. A. JENKINS and 1. F. TRAUB: A three-stage algorithm ]or real polynomials using quadratic iteration, SIAM J. Numer. Anal. 7, (1970), M. A. JENKINS and J. F. TRaun: Principles for testing polynomial zerofinding programs, ACM Trans. Math. Software 1, (1975), B. T. SMITH: ZERPOL, a zero finding algorithm /or polynomials using Laguerre's method, M. S. Thesis, Depertment of Computer Science, University of Toronto, (1967). ]. H. WXLKINSON: Rounding Errors in Algebraic Processes (1963), Prentice-Hall, Englewood Cliffs, N. J..